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Zhou et al. Intell Robot 2023;3(1):95-112 I http://dx.doi.org/10.20517/ir.2023.05 Page 109

60 50
AFOFTSMC AFOFTSMC
40 CSMC CSMC
FTSMC FTSMC
20 0
Torque (Nm) -20 0 Torque (Nm)
-40
-60 -50
-80
-100 -100
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Time(s) Time(s)
(a) Control inputs of hip joint (b) Control inputs of knee joint

Figure 4. AFOFTSMC control input comparison with CSMC and FTSMC. CSMC: conventional sliding mode control; FTSMC: fast terminal
sliding mode control; AFOFTSMC: adaptive fractional order fast terminal sliding mode controller.

0.035 0.06
AFOFTSMC AFOFTSMC
0.03 CSMC 0.05 CSMC
FTSMC 0.04 FTSMC
Tracking error (rad) 0.015 Tracking error (rad) 0.02 0
0.025
0.03
0.02
0.01
0.01
0.005
0 -0.01
-0.02
-0.005 -0.03
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Time(s) Time(s)
(a) Tracking errors of hip joint (b) Tracking errors of knee joint
Figure 5. Tracking errors comparison of CSMC, FTSMC, and AFOFTSMC in simulations. (a) errors comparison of the hip joint; (b) errors
comparison of the knee joint. CSMC: conventional sliding mode control; FTSMC: fast terminal sliding mode control; AFOFTSMC: adaptive
fractional order fast terminal sliding mode controller.

1 5
AFOFTSMC AFOFTSMC
4
CSMC CSMC
0.5
FTSMC FTSMC
3
0 2
s1(k) -0.5 1 0 s2(k)
-1 0.02 -1 0.1
0.01
-2 0
-1.5 0
-3
-0.01 -0.1
0.054 0.055 0.056 0.057 0.058 0.059 0.05 0.055 0.06 0.065
-2 -4
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
Time(s) Time(s)
(a) Sliding variables 1 ( ) (b) Sliding variables 2 ( )
Figure 6. Sliding variables for the three sliding surfaces.
To provide more quantitative proof, three indicators are compared in Table 1 to evaluate the performance of
three different control strategies. Mean absolute error (MAE) with = 1 Í | ( )|, maximum tracking
=1
q Í
2
error (MTE) with = | ( )|, root mean square error (RMSE) with = 1 [ ( )] .
=1
The results are shown in Table 1. The discrete adaptive fractional order fast terminal sliding mode control
algorithm proposed in this paper is applied to the lower limb exoskeleton robot to achieve better trajectory
tracking control. The generated RMSE and MAE are both minima, but the MTE is maximum, which is due to
overdrive caused by fast response speed.

To enhance SMC’s ability to resist external disturbances, we use fractional order sliding mode surfaces to accel-